You can read Eryxias online in Benjamin Jowett's translation or, if your prefer French, in Victor Cousin's translation.
(in order of appearance)
I haven't been able to find anything about him, but as 'Eryxias' is a name found in Critias's family, it is likely he was some kind of relative of Critias.
Critias was cousin to Plato's mother, according to Diogenes Laertius. He was a highly educated man, a poet and dramatist, very well respected in his day for his writings. However, when Athens fell at the end of the Peloponnesian War, the Spartans instituted an oligarchy, the Thirty Tyrants. Critias was one of the thirty; indeed, one of the most prominent. The Thirty Tyrants then initiated a reign of terror, killing anyone who might possibly threaten their regime or might support Thrasybulus and the other Athenian generals away from Athens who were working to restore Athenian democracy. He was killed in a battle against Thrasybulus's forces when they retook Athens. Socrates's association with him might have contributed to the jury's lack of sympathy when Socrates was on trial. Critias also appears in Charmides and Protagoras; the Critias of Timaeus and Critias is probably his grandfather. Xenophon portrays him as very much cynic out for his own gain, associating with Socrates because he thinks it is useful for gaining power.
Erasistratus is of notable family. His grandfather, also called Erasistratus (of Ceos), was an important physician, and was, indeed one of the most important anatomists of the ancient world, giving detailed descriptions of the brain as well as of the veins and arteries. His uncle Phaeax was part of an important embassy to Sicily in 422 BC to try to build alliances against Syracuse; according to this dialogue (but as far as I can tell only this dialogue) Erasistratus went with him. Erasistratus himself became one of the Thirty Tyrants.
Prodicus of Ceos
Prodicus was one of the greatest of the Sophists of ancient Greece; he is mentioned multiple times throughout the dialogues of Plato and Xenophon, and is a character in Protagoras. He is said to have been put to death for corrupting the youth, as Socrates was. Here he is not a character in the main dialogue, but a character in a dialogue-within-the-dialogue.
Unnamed young man
He is Prodicus's primary interlocutor in the dialogue-within-the-dialogue.
The Plot and The Thought
Socrates, who is narrating, is walking with Eryxias around the Stoa of Zeus Eleutherios when they meet Critias and Erasistratus. Erasistratus has just come back from the embassy to Sicily and Italy, after having spent the previous day walking from Megara. They sit down and Erasistratus makes some comments about Syracuse, noting that they've just sent an embassy to Athens and are probably trying to trick the Athenians. The Syracusan embassy happens to walk by and Erasistratus points out one of them, who is said to be the wealthiest man in all of Sicily and Italy. Socrates asks what his reputation is, and Erasistratus replies that he's also regarded as the wickedest man in Sicily and Italy. Socrates sees this as an opening for a discussion of virtue and wealth, and he will attempt to convince the others that wisdom is more valuable than other possessions, so the wise man is wealthier than the man with considerable property.
Eryxias jumps in and protests that no matter how wise a person is, if he lacks basic necessities, his wisdom is of no use. Socrates goes on to argue that what counts as wealth depends on what people can actually use. If you have a house full of gold and silver, you can only get even basic necessities if other people want the house or its contents; whereas if people wanted wisdom, the wise person would never have a problem getting basic necessities. Eryxias, annoyed, asks if he really thinks that he is wealthier than Callias, son of Hipponicus. Socrates criticizes him for thinking argument is just a game. Eryxias responds that they should be talking about whether it is good and bad to be wealthy. Socrates asks for his view on that question, and Eryxias responds that it is good to be wealthy.
Critias interrupts and asks if this is really his view. Critias will then argue against Eryxias that whether wealth is good depends on what it is used for; so that, for instance, someone who is using his wealth to commit adultery would be better off without the wealth. Critias outmaneuvers Eryxias easily so that "if it weren't for the embarrassment Eryxias was feeling in front of everyone there, he might very well have stood up and hit Critias" (397c). To defuse tension, Socrates gives us the dialogue-within-the-dialogue, telling about a discussion he participated in a few days ago on this very subject with Prodicus of Ceos. Prodicus had argued the same thing as Critias, that wealth is good for the good and bad for the bad. After reporting (somewhat obscurely) that dialogue, Socrates remarks that this seems to shed light on how people handle philosophy: Prodicus got thrown out of the gymnasium for making arguments like Critias, whereas everyone here agrees with Critias, so it seems that they determine the quality of arguments on the basis of the arguer's character rather than the arguments themselves.
Socrates asks Eryxias what he thinks wealth is, and Eryxias replies that it's the possession of lots of property; to which Socrates replies that this just requires that we consider what property is. If you had a large stash of cash that you couldn't possibly use, there's a sense in which you possess it, but it's actually worthless to you: you're not actually any wealthier for having it. So (in the relevant sense) it seems that everything useful is property. Eryxias counters that while everything that's property is useful, the reverse is not true: property is only one kind of useful thing. So what kind of useful thing could it be? Socrates proposes that they look at what would have to be removed in order to remove our need for property, and suggests as a possible answer (based on Eryxias's prior argument about why wisdom does not make Socrates wealthier than Callias), bodily needs, so that without bodily needs property wouldn't exist at all. Eryxias is getting confused at this point, but agrees. But if this is the case, gold, silver, and the like are only property (in the sense relevant to wealth) if they are useful for meeting bodily needs; otherwise, they aren't property. Eryxias says that gold and silver obviously are property; we do in fact find them useful for living.
At this point Socrates notes that lots of people sell their skills to get what they need for living; which means that they are property that contributes to wealth in exactly the same way gold and silver are. Further, even if you have gold and silver, they can only be useful for getting what the body needs if you know how to use them, and likewise, all that you need to make things useful to someone is to give him the right kind of knowledge. So whether or not something is wealth depends on whether you have the knowledge and wisdom required to use it. Socrates notes that Critias does not agree, and Critias says that Socrates's position is crazy, but asks Socrates to continue his argument; Socrates remarks that Critias enjoys listening to arguments like someone enjoys listening to a rhapsode reciting Homer, not believing any of it is true.
The argument about the relation between property and usefulness continues a little longer, but Socrates, noting that Critias is not being persuaded, changes his tack. Are we better off when we are healthy or when we are sick? Critias gives the obvious answer, and Socrates gets him to concede that the reason the sick person is worse off is that he has a greater need for all kinds of things -- and so, likewise, the person with the gambling or drinking or eating habit. But the person who has the most useful things for meeting the needs of the body seems to need all those useful things, otherwise they wouldn't be useful. So, Socrates ends the dialogue, somewhat abruptly, with a suggestion:
According to this argument, at least, it appears that those who have a lot of property must also need many of the things required to take care of the body, since property was seen as useful for this purpose. So the wealthiest people would necessarily appear to us to be in the worst condition, since they are in need of the greatest number of these things. (406a)
One of the noticeable things about the dialogue is that there seems to be an undercurrent here: Socrates comments negatively several times on how people treat arguments -- like a game, like a rhapsody, as something to be judged entirely in terms of the character of the arguer. In other words, Socrates's interlocutors do not find the arguments to be very useful; but this -- one might conclude from the claims made in the argument itself -- is because they do not have the appropriate knowledge and wisdom to find it useful. In this sense, wealth might almost be considered a secondary topic in the dialogue: what is really at issue is the opposition between those who have the wealth of wisdom and those without the wisdom to recognize that wisdom is true wealth, between the philosophers who seek wisdom rather than gold and silver, and those who are dominated by uncritical acceptance of commonplaces about what really makes one wealthy.
* The Stoa, or Porch, of Zeus Eleutherios was a walking & gathering space that was a monument from the Persian War. 'Eleutherios' means 'freedom', so it might possibly be an ironic setting, given that the dialogue ends with a discussion of lives dominated by needs.
* The Peloponnesian War led Athens to start taking an interest in Sicily. Syracuse, worried about Athenian power, tried to get all the Sicilians to agree to exclude foreigners from the island. The embassy led by Phaeax in 422 BC, from which Erasistratus is returning, foiled this plan by making alliances with Sicilian cities that were suspicious of Syracuse. The result would set up conditions that would later contribute to one of Athens's worst disasters in the war, the Sicilian Expedition, in which Athens set out to conquer Sicily with a massive expeditionary force that was in the end almost entirely destroyed. One way to read this setting is to see it as an indictment of Athens, greedy for wealth of "things that make you wealthy--slaves, horses, gold, and silver" (392d), and unable, in the persons of Erasistratus, Eryxias, and Critias, to grasp that these things are not what really matter -- a greed and folly for which Athens would pay very dearly.
* Megara is somewhere around twenty miles from Athens, and was on the Spartan side in the Peloponnesian war.
* Callias, son of Hipponicus, was one of the wealthiest people in Athens, and a patron of sophists. Plato's Protagoras takes place at his house.
Quotations from Mark Joyal's translation in Plato, Complete Works, Cooper & Hutchinson, eds., pp. 1718-1733.